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In this article we explain the theory of rigid residue complexes in commutative algebra and algebraic geometry, summarizing the background, recent results and anticipated future results. Unlike all previous approaches to Grothendiec…
Depending on its geometry, a spherical shell may exist in one of two stable states without the application of any external force: there are two `self-equilibrated' states, one natural and the other inside out (or `everted'). Though this is…
A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle…
Hierarchical structures are very common in Nature, but only recently have they been systematically studied in materials physics, in order to understand the specific effects they can have on the mechanical properties of various systems.…
This paper investigates the rigidity of bordered polyhedral surfaces. Using the variational principle, we show that bordered polyhedral surfaces are determined by boundary value and discrete curvatures on the interior edges. As a corollary,…
We reveal significant qualitative differences in the rigidity transition of three types of disordered network materials: randomly diluted spring networks, jammed sphere packings, and stress-relieved networks that are diluted using a…
Semiconductor $p^+ - p^- - n - p^+ - n^{++}$ structures with large junction and contact areas are treated as 1 \times 2-dimensional active media, in which self-organized pattern formation can be expected. The local bistable behavior of the…
The paper develops the stiffness relationship between the movements and forces among a system of discrete interacting grains. The approach is similar to that used in structural analysis, but the stiffness matrix of granular material is…
Marginal stability is the notion that stability is achieved, but only barely so. This property constrains the ensemble of configurations explored at low temperature in a variety of systems, including spin, electron and structural glasses. A…
The shape of semiflexible polymer rings is studied over their whole range of flexibility. Investigating the joint distribution of asphericity and nature of asphericity as well as their respective averages we find two distinct shape regimes…
A two dimensional amorphous material is modeled as an assembly of mesoscopic elemental pieces coupled together to form an elastically coherent structure. Plasticity is introduced as the existence of different minima in the energy landscape…
A topological interlocking assembly consists of rigid blocks together with a fixed frame, such that any subset of blocks is kinematically constrained and therefore cannot be removed from the assembly. In this paper we pursue a modular…
In many interesting physical settings, such as the vulcanization of rubber, the introduction of permanent random constraints between the constituents of a homogeneous fluid can cause a phase transition to a random solid state. In this…
Despite the fundamental importance of solid--solid transitions for metallurgy, ceramics, earth science, reconfigurable materials, and colloidal matter, the details of how materials transform between two solid structures are poorly…
Flexible slender structures such as rods, ribbons, plates, and shells exhibit extreme nonlinear responses bending, twisting, buckling, wrinkling, and self contact, that defy conventional simulation frameworks. Discrete Differential Geometry…
The application of stress to multiphase solid-liquid systems often results in morphological instabilities. Here we propose a solid-solid phase transformation model for roughening instability in the interface between two porous materials…
Plasticity in amorphous materials, such as glasses, colloids, or granular materials, is mediated by local rearrangements called "soft spots". Experiments and simulations have shown that soft spots are two-state entities interacting via…
We numerically study the static structure and the mechanical response of two-dimensional granular piles. The piles consist of polydisperse disks with and without friction. Special attention is paid for the rigid grain limit by examining the…
A two-dimensional direction-length framework $(G,p)$ consists of a multigraph $G=(V;D,L)$ whose edge set is formed of "direction" edges $D$ and "length" edges $L$, and a realisation $p$ of this graph in the plane. The edges of the framework…
Adding perforations to a continuum sheet allows new modes of deformation, and thus modifies its elastic behavior. The failure behavior of such a perforated sheet is explored, using a model experimental system: a material containing a…